Phase Transition Signals in Sparse Longitudinal Cognitive Trajectories: A Regime-Switching Framework for Pre-Diagnostic Instability in Alzheimer's Disease (ADNI)

Phase Transition Signals in Sparse Longitudinal Cognitive Trajectories: A Regime-Switching Framework for Pre-Diagnostic Instability in Alzheimer's Disease Aamish Ahmad * *School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India Email: aamish. ahmad99@gmail. com Abstract Alzheimer's Disease (AD) progresses as a continuous biological process, yet clinical diagnosis proceeds in discrete stages, i. e. , Cognitively Normal (CN), Mild Cognitive Impairment (MCI), and AD defined by threshold crossings on standardised assessments. This mismatch creates an observational gap: a period during which the cognitive system is destabilising, but no diagnostic threshold has been crossed. This work examines whether the pre-diagnostic transition regime produces measurable signals in routine longitudinal clinical data. The central hypothesis is that the CN-to-MCI transition is better characterised as a latent regime-switching process than as a discrete event. Under this framing, the transition period is expected to exhibit increased variance, weakened signal consistency, and divergence across cognitive instruments. Using longitudinal assessment data from ADNI (MMSE, ADAS-Cog, CDR-SB, FAQ), per-patient trajectory features are organised into four system classes, such as state, trend, instability, and cross -instrument misalignment, and computed over a pre-registered analysis window of 36 months prior to conversion. This is not a predictive model and does not produce classification outputs. It is a signal characterisation study asking whether instability estimators derived from sparse clinical time series show statistically detectable distributional differences between converter and stable groups. No experimental results are reported. This document constitutes the theoretical framing, methodological design, and pre-registered analytical specifications of the study. 1. Introduction Clinical diagnosis of Alzheimer's Disease captures a system that has already undergone substantial change. The transition from cognitively normal status to MCI represents not an instantaneous failure but a progressive loss of stability in an underlying biological system. The standard analytical framework treats diagnosis as a point in time, a boundary crossed, after which the patient is reclassified. This conflation of biological transition with an administrative label is a known limitation of longitudinal cohort studies in neurodegeneration. The theoretical motivation draws from dynamical systems theory, specifically Critical Slowing Down (CSD): systems approaching a regime transition exhibit increased variance and slower recovery from perturbation. If cognitive decline follows this pattern, the period preceding MCI diagnosis should be statistically distinguishable from stable CN behaviour not by score level alone, but by the character of trajectory dynamics. This study operationalises that claim using ADNI longitudinal data. Rather than modelling prediction or classification, it asks: do the trajectory dynamics of converters and stable subjects differ in the pre-diagnostic window, and in what direction? 2. Theoretical Framework The cognitive system of a patient is governed by a latent hidden state Z (t) that takes values in CN, MCI, AD. This state is not directly observable. What is observed are noisy clinical measurements MMSE, ADAS-Cog, CDR-SB, and FAQ, which constitute a noisy projection of the true system state: X (t) = f (Z (t) ) + e (t) where e (t) represents measurement noise and within-visit variability. A regime is a period of statistically consistent system behaviour. The CN regime exhibits slow, regular drifts with low residual variance. The MCI regime shows a new, lower-performing stable baseline. The transition regime, the object of this study, is characterised by elevated variance, non-monotonic fluctuations, and cross-instrument divergence as the hidden state probabilistically shifts from CN to MCI. A regime shift is not a change in score level. It is a change in the statistical character of the trajectory: the system loses stability properties before settling into a new regime. Two patients may show identical scores at a given visit while exhibiting fundamentally different trajectory dynamics. 3. Data and Observational Setting This study uses the ADNI ADNIMERGE merged longitudinal table. Variables used: DX (CN/MCI/AD), MMSE, ADAS-Cog, CDR-SB, and FAQ. These instruments were selected because they are longitudinal, measured at every visit, semantically related, and interpretable without additional processing. Static or irregularly collected variables, such as MRI volumetrics, PET biomarkers, CSF markers, and genetics, are excluded. Structural Constraints • Sparse trajectories: 3 to 7 visits per patient, typical • Irregular visit intervals • Missing values, particularly at later time points • No ground-truth transition onset label, only DX conversion is observable • DX label lags biological transition by an uncertain and variable amount 4. Feature Construction All features are computed per subject and normalised relative to the subject baseline, where applicable. Features are organised into four system classes. State Features mmseₗast, adasₗast, and cdrₗast are the most recent observed values of each instrument. They establish where the system currently is, providing the baseline level against which all change is measured. Trend Features mmseₛlope, adasₛlope, mmsedelta direction and rate of system movement. Slope is estimated by linear regression over available visits. Delta records the total displacement from the first to the last visit. A system approaching a phase transition may not change its trajectory direction as stability changes. Instability Features (Primary Signal) Instability is defined as the residual component of trajectory variance after removing the linear trend: Total variance = trend component + residual component The residual component (mmseᵣesidualᵥar) represents variance unexplained by directional drift. Under CSD, this is predicted to increase during a transition regime. A second feature, reversalcount, records direction changes in the MMSE trajectory, capturing non-monotonic instability distinct from variance-based estimation. Misalignment Features In a stable regime, MMSE and ADAS-Cog should co-move. If decline begins unevenly, normalised slopes will diverge before either instrument crosses a diagnostic threshold: crossₛignaldiff = |MMSEₛlopeₙormalized - ADASₛlopeₙormalized| Both slopes are normalised prior to subtraction to correct for different scales and directional conventions (MMSE: higher is better; ADAS-Cog: lower is better). Feature Summary Class Feature Definition State mmseₗast Most recent MMSE value State adasₗast Most recent ADAS-Cog value State cdrₗast Most recent CDR-SB value Trend mmseₛlope Linear slope of MMSE over visits Trend adasₛlope Linear slope of ADAS-Cog over visits Trend mmsedelta Last minus first MMSE value Instability mmseᵣesidualᵥar Variance after linear detrending Instability reversalcount Direction changes in MMSE trajectory Misalignment crossₛignaldiff |MMSEₛlopeₙorm - ADASₛlopeₙorm| 5. Method The analytical pipeline proceeds in seven steps. 1. Extract CN baseline subjects with 3 or more valid MMSE visits from ADNIMERGE. 2. Identify converter cohort (CN to MCI) and stable cohort (CN throughout all visits). 3. Define tc = date of first MCI diagnosis per converter subject. 4. Re-index converter visits: tau = t - tc. Pre-registered window: tau in -36 months, 0. Fixed prior to data access. 5. Compute the full feature vector per subject. 6. Compare distributions using Mann-Whitney U tests; report Cohen's d effect sizes. 7. Conduct failure mode analysis: document the proportion of subjects for whom each estimator is undefined or unreliable. 6. What This Work Is Not • Not a predictive model, no probability of conversion is estimated • Not a classification system, no train-test split, no AUC, no accuracy metric • Not a clinical diagnostic tool; no medical claims are made • Not deep learning-based 3 to 7 visit trajectories make neural approaches inappropriate • No results reported. This document is a pre-experiment specification 7. Hypotheses and Falsification If the hypothesis holds: • mmseᵣesidualᵥar is higher in converters than in stable CN subjects • reversalcount is elevated in converters • crossₛignaldiff increases in converters as tau approaches 0, separating groups earlier than state features Falsification Condition (pre-stated, fixed prior to data access): Falsified if Mann-Whitney U yields p > 0. 05 AND Cohen's d < 0. 2 for all instability features within tau in -36 months, 0. Negative results will be reported in full and will not be reframed post-hoc. 8. Current Status and Next Steps At the time this document was prepared, the theoretical framework was fully specified, feature definitions were complete, and the pre-registered analytical window was locked. No data extraction has been executed, and no experimental results exist. Execution Roadmap 1. Data extraction and converter group identification from ADNIMERGE. 2. End-to-end feature computation pipeline implementation. 3. Statistical testing per the protocol above. 4. Failure mode quantification proportion of subjects for whom each estimator is computable and reliable. 5. Results to be reported in a subsequent version or companion paper.